MinT - Best Known (211−123, 211, s)-Nets in Base 4

Best Known (211−123, 211, s)-Nets in Base 4

(211−123, 211, 104)-Net over F₄ — Constructive and digital

Digital (88, 211, 104)-net over F₄, using

t-expansion [i] based on digital (73, 211, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(211−123, 211, 129)-Net over F₄ — Digital

Digital (88, 211, 129)-net over F₄, using

t-expansion [i] based on digital (81, 211, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(211−123, 211, 879)-Net in Base 4 — Upper bound on s

There is no (88, 211, 880)-net in base 4, because

1 times m-reduction [i] would yield (88, 210, 880)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 864139 096767 210641 801259 019085 956541 395236 964584 669254 708297 601244 300058 694639 203758 024418 014403 192891 701218 369815 304130 661276 > 4²¹⁰ [i]